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A spokesman for TfL said the refund would amount to "a few thousand pounds" but denied taxpayers would be out of pocket.

"A small number of members had not docked their bicycles correctly when they finished their journeys," said the spokesman. "As a measure of goodwill we waived the journey fares for everyone who used the bikes on launch day. The vast majority of those were making free trips of under 30 minutes so the refund is mostly revenue we were not expecting to have in the first place.

"When people re-dock hire cycles, they need to ensure that the light on the docking point turns green and that their cycle clicks into place."

Users on internet cycle forums complained of being unable to dock their bikes in time to avoid charges and said some docking locations were not open.

One cyclist told the BBC: "Unfortunately I couldn't pull the bike out. [But] the computer's reporting that I've been cycling for 11 hours, 11 minutes and 11 seconds."

The TfL spokesman said none of the Canadian bikes, which cost "£800 or £900 each" to build, had yet been stolen.

TfL is spending £114m over five years on the project, which it eventually expects to recoup in charges, while sponsor Barclays is contributing £25m over the same period.

Mr Johnson, who describes himself as cycling like a "very elderly French onion seller", admitted he is expecting "delirious" criticism of the scheme.